Maximal Normalized Steklov Eigenvalues

Definitions and mathematical statement

The supplied record asks an extremal question but does not specify the underlying manifold or the precise class of admissible geometries. The following gives a conditional formulation using the standard Steklov problem; these geometric hypotheses are needed to interpret the question and are not fully specified by the record.

Let $M$ be a specified compact smooth manifold of dimension $n+1$, where $n\geq 1$, with nonempty smooth boundary $\Sigma=\partial M$. For a smooth Riemannian metric $g$ on $M$, a Steklov eigenvalue is a number $\sigma$ for which there exists a nonzero smooth function $u$ satisfying \[ \Delta_g u=0\quad\text{in }M,\qquad \partial_{\nu_g}u=\sigma u\quad\text{on }\Sigma, \] where $\nu_g$ is the outward unit normal. Under the convention of indexing from zero, write the eigenvalues, counted with multiplicity, as \[ 0=\sigma_0(M,g)\leq\sigma_1(M,g)\leq\sigma_2(M,g)\leq\cdots. \] This convention is explicit here because the supplied record does not specify its indexing convention or connectedness assumptions. Let $|\Sigma|_g$ denote the $n$-dimensional Riemannian measure of the boundary, and let $\mathcal G(M)$ denote all smooth Riemannian metrics on this specified $M$.

For a fixed admissible index $k$, the optimization question, if the intended domain is $\mathcal G(M)$, is to determine \[ S_k(M)=\sup_{g\in\mathcal G(M)} \sigma_k(M,g)|\Sigma|_g^{1/n} =\sup_{\substack{g\in\mathcal G(M)\\|\Sigma|_g=1}} \sigma_k(M,g). \] The equality expresses the boundary-measure normalization by constant rescaling of the metric. The supremum is understood in $[0,+\infty]$. The source asks for a maximum; that wording additionally requires determining whether a finite supremum is attained by a smooth metric.

The record does not establish whether $M$, its dimension, its topology, or any additional geometric restrictions are fixed. Thus $S_k(M)$ is a precise conditional interpretation, not an assertion that this particular optimization class is the intended one. No single numerical extremal target or attainment claim can be extracted without resolving these qualifications.

Short English statement

For a fixed eigenvalue index, ask how large a Steklov eigenvalue can become after normalizing the boundary measure. On a specified manifold this is an optimization over smooth metrics, with a separate question of whether the supremum is attained. The supplied record leaves the manifold, admissible geometry, and indexing convention unresolved.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30002113$, exact source problem code catalogue entry, title Maximal Normalized Steklov Eigenvalues; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The proposed year is recorded as 2012.
• Supplied report citation: Geometric Aspects of Spectral Theory (2013), Oberwolfach Reports, pp. 2013--2076, DOI 10.4171/owr/2012/33. https://doi.org/10.4171/owr/2012/33. The citation's title-year designation is retained as supplied.
• Supplied literature source, labelled "OWR report": https://ems.press/content/serial-article-files/46403.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and says that the extracted record is only a heading, does not identify one precise claim, and that the report gives bounds and extremal examples without a complete general answer. This is supplied dated curation, not an independently verified literature assessment.
• The registry also labels the statement corrected_verified and reports that the official report confirmed the complete optimization question after repairs to the index and normalization notation. This supplied verification claim has not been independently checked here; its tension with the heading-only literature assessment remains unresolved.